Graphing Ellipses Worksheet: Center, Axes, and Foci

Algebra and Function Practice

An ellipse is the set of points whose distances to two fixed foci sum to a constant. Its standard equation reveals the center, the lengths of the major and minor axes, and the locations of the foci. This lesson walks through reading those features from an equation and using them to sketch an accurate graph. You will practice marking the center, vertices, co-vertices, and foci, then drawing a smooth closed curve through the vertices and co-vertices. The convention used here is that the major axis is the longer axis, the minor axis is the shorter one, and foci lie on the major axis. All numerical examples are hypothetical and chosen for clean arithmetic; measure your own grid or tools if you adapt the tasks.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Reading the Standard Equation

The standard form for a horizontal major axis is (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1 with a > b > 0. The center is (h, k). The major axis is horizontal and has length 2a; the minor axis is vertical and has length 2b. For a vertical major axis, the denominators swap: (x - h)^2 / b^2 + (y - k)^2 / a^2 = 1 with a > b > 0. Always compare denominators: the larger denominator sits under the variable whose axis is major.

Vertices are the endpoints of the major axis, at distance a from the center along the major direction. Co-vertices are the endpoints of the minor axis, at distance b from the center along the minor direction. The foci lie on the major axis at distance c from the center, where c^2 = a^2 - b^2. Because a > b, c is real and c < a, so foci stay inside the ellipse. Label all points clearly and keep units consistent with the grid.

1. Identify h and k from the parentheses to locate the center.
2. Compare denominators to decide whether the major axis is horizontal or vertical.
3. Compute a and b by taking square roots of the larger and smaller denominators.
4. Mark vertices at distance a from the center along the major axis.
5. Mark co-vertices at distance b from the center along the minor axis.
6. Compute c^2 = a^2 - b^2 and mark foci at distance c from the center on the major axis.
7. Draw a smooth closed curve through the four vertices and co-vertices.

Plotting Points and Sketching the Curve

Work on a labeled coordinate grid. Plot the center first, then count grid units along the major and minor directions. For example, if a = 5 and the major axis is horizontal, vertices are 5 units left and right of the center. If b = 3, co-vertices are 3 units up and down. Foci are placed using c, not a or b. A common error is to place foci at the co-vertices; instead, foci always lie on the major axis.

When sketching, draw lightly through the four axis endpoints and adjust so the curve is symmetric about both axes through the center. The ellipse should pass through all four endpoints and bulge smoothly between them. Check symmetry by confirming that opposite points are equidistant from the center. If your grid uses a different scale, state the scale and keep all distances in the same units.

1. Plot and label the center (h, k).
2. Count a units along the major axis and mark both vertices.
3. Count b units along the minor axis and mark both co-vertices.
4. Count c units along the major axis from the center and mark both foci.
5. Sketch a smooth closed curve through the four axis endpoints.
6. Verify symmetry and that foci lie inside the curve on the major axis.

Checking Your Graph

A correct graph satisfies three checks. First, the center is exactly at (h, k). Second, the distances from the center to the vertices equal a and to the co-vertices equal b. Third, the foci are on the major axis at distance c from the center with c^2 = a^2 - b^2. You can also test one point: substitute a vertex coordinate into the equation and confirm the result is 1.

Use arithmetic that is easy to verify. For instance, if a^2 = 25 and b^2 = 9, then a = 5, b = 3, and c^2 = 16 so c = 4. The foci are 4 units from the center along the major axis. Keep all lengths in grid units and avoid mixing units. If you choose different numerical values for your own practice, label them as hypothetical examples and recompute c from a^2 - b^2.

1. Confirm center coordinates match h and k.
2. Measure vertex distances and compare to a.
3. Measure co-vertex distances and compare to b.
4. Confirm foci lie on the major axis and their distance equals c.
5. Check that c^2 = a^2 - b^2 using the same units.

Worked example

Graph the ellipse (x - 2)^2 / 25 + (y + 1)^2 / 9 = 1. Identify the center, major and minor axes, vertices, co-vertices, and foci.

Four practice exercises

Common mistakes

• Swapping the major and minor axes by ignoring which denominator is larger. Compare denominators first; the larger denominator belongs to the major axis direction.
• Placing foci on the minor axis or at the co-vertices. Foci always lie on the major axis at distance c from the center, where c^2 = a^2 - b^2.
• Using a and b directly as coordinates without adding or subtracting from the center. Vertices and co-vertices are measured from the center (h, k), not from the origin unless the center is the origin.
• Computing c as a + b or as the difference of denominators. Use c^2 = a^2 - b^2 with a and b as lengths, then take the positive square root.

Self-check rubric

• Center correctly identified and plotted at (h, k).
• Major axis direction correctly chosen by comparing denominators.
• Values of a and b correctly computed as square roots of denominators.
• Vertices plotted at distance a from center along major axis.
• Co-vertices plotted at distance b from center along minor axis.
• Foci plotted on major axis at distance c with c^2 = a^2 - b^2.
• Curve is smooth, closed, and passes through all four axis endpoints.
• Labels and units are consistent and readable.

Questions and answers